Hydrodynamic and Stochastic Limits of Space-Time Hawkes Processes

Source abstract

We study the large-scale behavior of critical and nearly critical space-time Hawkes processes initialized by a prescribed Poisson field of past events. The density of this initial field separates three macroscopic regimes. In the high-density regime, fluctuations average out and the event measure converges to a hydrodynamic limit which is a deterministic diffusion profile. At the fluctuation threshold, random branching remains visible and produces a measure-valued stochastic limit. Below the threshold, the rescaled event measure vanishes locally even though its mean generally remains nontrivial, with rare large clusters accounting for the surviving mean. For reproduction delays with finite mean, the critical limits are described by heat flow and the occupation measure of a branching diffusion. A nearly critical perturbation introduces linear decay without changing the three-regime structure. Heavy-tailed delays lead to a different balance between time and space: the mean follows a fractional diffusion law, the stochastic limit is a superprocess occupation measure on time-space, characterized by a nonlinear fractional equation, and the past persists as a temporal memory term. We also determine how the scaling of the Poisson past must be matched to the observation scales and show that, in one spatial dimension, a shorter observation window recovers stochastic behavior hidden in the sparse regime. The results allow bounded initial profiles with possibly infinite total mass and describe the limits directly at the level of locally finite event measures.